Did the Greeks Discover the Irrationals?
Philip Hugly and Charles Sayward

Abstract

A popular view is that the great discovery of Pythagoras was that there are irrational numbers, e.g., the positive square root of two. Against this it
is argued that mathematics and geometry, together with their applications, do not show that there are irrational numbers or compel assent to that
proposition.

Ok, that is two sentences from the link you sent me. Do you understand the arguements against irrational numbers? If so can you explain the arguments
against them to me so I don't have to spend $34 to buy the article? I mean, since you brought it up and used it in the context that things like this
are destroying the world I figure you must have a pretty good understanding of the ideas you suscribe to.

So you don't have a post-secondary library in your city? You can read the article for free in their library -- the journal Philosophy will be
there.

As for the argument -- it's the difference between distance and length. Length is geometric whereas difference is arithmetic. So the Pythagorean
Theorem is proof by contradiction -- it proves that the LENGTH is not rational but it does not prove that the distance is the square root of two. The
proof relies on a "divide and average" process of arithmetic which is never ending. To state that this arithmetic process of distance is the same
as the geometric measurement of length is not logically accurate. Granted it is mathematically precise -- so that 3/2 squared is 9/4 divided by 2 is
9/8 doubled is 10/8 which is 5/4 and 9/8 cubed? Approximates the square root of two -- that's the original proof from music theory. So 5/4 was the
cube root of two and 9/8 cubed is the square root of two and the 12th root of two is the equal-tempered tuning measurement for the 12 notes of the
music scale. But the original source is 3/2 as the perfect 5th -- again 3/2 squared is 9/4 and halved is 9/8 -- but 3/2 and 2/1 do not line up -- in
other words the 12th root of two as 12 notes equals the octave but the 12 fifths as 3/2 squared do not equal the octave equivalence -- 7 octaves and
12 fifths do not line up.

This is, again, because the arithmetic as distance is not the same as the geometry as length -- so the arithmetic is called the "comma of
Pythagoras" -- and it's covered up since the octave doubled as 4 is also the octave squared as 4 -- and then you take the square root and get rid of
the noncommutative arithmetic.

Bertrand Russell stated that the real numbers are a "convenient fiction" because of Cantor's proof about negative sets -- you can not prove that
the irrational numbers are a truly infinite set -- only that they are larger than the rational numbers.

So then the problem is that if you believe that for any infinite set there's a set that's even larger, what happens if you apply this to the
universal set, the set of everything? The problem is that by definition the set of everything has everything, and this method supposedly would give
you a larger set, which is the set of all subsets of everything. So there's got to be a problem, and the problem was noticed by Bertrand Russell.
Bertrand Russell Cantor I think may have noticed it, but Bertrand Russell went around telling everyone about it, giving the bad news to everyone! ---
At least Gödel attributes to Russell the recognition that there was a serious crisis. The disaster that Russell noticed in this proof of Cantor's
was the set of all sets that are not members of themselves, that turns out to be the key step in the proof. And the set of all sets that aren't
members of themselves sounds like a reasonable way to define a set, but if you ask if it's inside itself or not, whatever you assume you get the
opposite, it's a contradiction, it's like saying this statement is false. The set of all sets that are not members of themselves is contained in
itself if and only if it's not contained in itself.

Just to bring this back to topic -- that link I gave is by a professor who promotes "Metabiology" -- replacing Nature with synthetic ecology!!

The simplicity of the discovery makes it "a beautiful result," Nemenman says. "We hope that this theoretical finding will also have practical
applications."

I'll be more impressed if I see practical applications of it instead of just looking at a theoretical result.

You can over-simplify a problem, and you can over-complicate a problem. What we engineers are trained to do is look at an appropriate level of
complexity of a problem to come up with an effective solution to the problem (in the real world, not the theoretical world). Sometimes the simplified
approaches work great, sometimes if you oversimplify and don't examine the complexities it can be a disaster, like the Tacoma Narrows bridge film
they seem to show every engineer, or should.

Personally, I do not find biology all that complicated. Like everything else, there are basic rules that can be followed. Everything is a variant of
those rules. Everything really IS quite simple, the problem is finding that keystone that shows us just how simple it is.

However, you can also look at things from the "forest" rather than the "trees" perspective and see just how complex a simple rule can actually
become.

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